Foundation Qrft Higgs Potential From Recognition Vacuum Higgs Potential

The potential as a cost

The standard model's Higgs potential is usually written V(H) = −μ²|H|² + λ|H|⁴, a quartic curve with a minimum at the electroweak vacuum expectation value v/√2 ≈ 174 GeV. That curve is the energy cost of the Higgs field sitting away from its preferred value. The Recognition Science declaration higgsPotential takes that same cost picture and gives it a different source: the cost is not put in by hand but is forced by the framework's central result, the J-cost function J(r) = ½(r + r⁻¹) − 1.

In this account, the potential is the J-cost evaluated on the ratio r = |H| / (v/√2), so the field's distance from its vacuum value is measured in units of the vacuum value itself. The declaration higgsPotential is simply a definition: it sets V_RS(r) = J(r). The content lies in what the framework proves about that definition. The vacuum has zero potential, J(1) = 0. The potential is symmetric about the vacuum, J(r) = J(r⁻¹), so moving the field above or below the vacuum by the same factor costs the same energy. The potential is never negative for positive r. And the vacuum is the unique minimum: J(r) = 0 if and only if r = 1. These four properties are collected in a certificate structure, higgsPotentialCert, and each is a proved theorem in the machine-checked library of formal theorems, not an assumption.

What the declaration does not do is derive the Higgs mass or the value of v from first principles. The mass squared is proportional to J''(1) = 1 in the framework's units, which is a calibration statement, a choice of scale, not a prediction of 125 GeV. The identification of the potential's argument with the physical field ratio is a modeling choice, not a theorem. The framework contributes a structural story: the Higgs VEV is the unique cost minimum, and electroweak symmetry breaking is the selection of r = 1 as the ground state. That is an interpretation of the standard mechanism, not a replacement for it.

The practical consequence is a compact restatement of a familiar object. The Higgs potential, one of the standard model's central pieces, is recovered as a special case of the forced cost function. The vacuum's uniqueness and stability are not additional assumptions but consequences of the cost function's shape. What remains open is whether the identification of the potential with J-cost is physically the right one, a question the framework's formal results do not settle.

/-- Higgs potential = J-cost on the field ratio. -/
def higgsPotential (r : ℝ) : ℝ := Jcost r

/-- The vacuum has zero potential. -/
theorem vacuum_zero_potential : higgsPotential 1 = 0 := Jcost_unit0

/-- The potential is symmetric about the vacuum. -/
theorem higgs_symmetric {r : ℝ} (hr : 0 < r) :
    higgsPotential r = higgsPotential r⁻¹ := Jcost_symm hr

/-- The vacuum is the unique minimum. -/
theorem higgs_unique_minimum {r : ℝ} (hr : 0 < r) :
    higgsPotential r = 0 ↔ r = 1 := by
  unfold higgsPotential
  constructor
  · intro h
    by_contra hne
    exact absurd h (ne_of_gt (Jcost_pos_of_ne_one r hr hne))
  · rintro rfl; exact Jcost_unit0

What this page does not claim

The Higgs mass of 125 GeV is derived from the framework's constants. The identification of the potential's argument with the physical field ratio is a proved theorem rather than a modeling choice. The framework's results replace the standard model's mechanism of electroweak symmetry breaking.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/QRFT/HiggsPotentialFromRecognitionVacuum.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: